Multiple Higgs-portal and gauge-kinetic mixings

Abstract

We develop a phenomenological formalism for mixing effects between the Standard Model and hidden-sector fields, motivated by dark matter in the Universe as well as string theories. The scheme includes multiple Higgs-portal interactions in the scalar sector as well as multiple gauge-kinetic mixings in the abelian gauge sector. While some of the mixing effects can be cast in closed form, other elements can be controlled analytically only by means of perturbative expansions in the ratio of standard scales over large hidden scales. Higgs and vector-boson masses and mixings are illustrated numerically for characteristic processes.

Appendices

Appendix A: Mixing in the hidden sector

If quartic [bi-bilinear] mixing terms in the hidden sector are included, the Higgs potential is generalized to

$$\begin{aligned} {\mathcal{V}}_{\mathcal{H}} =& \sum_{i=0}^n \bigl[ \mu^2_i |S_i|^2 + \lambda_i |S_i|^4 \bigr] +\frac{1}{2} \sum_{i \neq j=0}^n \eta_{ij} |S_i|^2 |S_j|^2 \\ =& \bigl[\mu^{2T} |S|^2 + |S|^{2T} \lambda |S|^2 \bigr] +\frac{1}{2} |S|^{2T} \eta |S|^2 \end{aligned}$$

(A.1)

in obvious vector/matrix notation in the second row. The index i=0 represents the Higgs field in the SM sector, i.e. S ₀=ϕ and η _0j =η _j [j=1,…,n] etc., while indices j≥1 refer to hidden-sector scalar fields.

The visible and hidden components v of the vacuum Higgs fields are defined by the vanishing of the derivative of the Higgs potential,

$$\begin{aligned} v^2 =& - \biggl\{ \lambda + \frac{1}{2} \eta \biggr\} ^{-1} \mu^2 \\ \simeq& - \biggl\{ \lambda^{-1} - \frac{1}{2} \lambda^{-1} \eta \lambda^{-1} \biggr\} \mu^2 \end{aligned}$$

(A.2)

for small off-diagonal mixing parameters η and ${\mathcal{O}}(1)$ diagonal parameters λ.

The term bilinear in the physical fields defines the masses of the Higgs particles,

(A.3)

Restricting the n×n symmetric mass matrix $M^{2}_{c}$ to the components of the hidden sector in the notation of Eq. (1.7), the matrix can be diagonalized by an orthogonal transformation ${\mathcal{O}}_{c}$, modifying subsequently the phenomenological mixing vector X in Eq. (1.8):

$$ \begin{aligned} &M^2_c \to M^2_{c/\operatorname{diag}} = {\mathcal{O}}_c M^2_c {\mathcal{O}}_c^T, \\ &X \to {\mathcal{O}}_c X . \end{aligned} $$

(A.4)

Notice that the off-diagonal ${\mathcal{O}}_{c}$ mixing elements change X only to higher order so that the portal interactions between visible and hidden sector fields are essentially not affected by the quartic mixing in the hidden sector.

Finally, the self-interactions among the Higgs fields, SM and hidden, can be derived from the potential

$$ {\mathcal{V}}_{\mathcal{H}/\mathrm{self}} = \biggl\{ \lambda + \frac{1}{2} \eta \biggr\} _{ij} \biggl[ v_i H_{ic} H^2_{jc} + \frac{1}{4} H^2_{ic} H^2_{jc} \biggr] $$

(A.5)

in terms of the physical Higgs fields in the current [c] representation.

Appendix B: Dyadic matrix

It is straightforward to diagonalize the n×n dyadic matrix D formed by a n-dimensional column vector x=(x ₁,…,x _n )^T and its transpose x ^T as

$$ D_{ij} = \bigl(x x^T\bigr)_{ij} = x_i x_j . $$

(B.1)

Making use of the rules for calculating determinants, one eigenvalue emerges as positive and the other (n−1) eigenvalues as zero:

$$\begin{aligned} d_1 = \sum^n_{i=1} x^2_i\quad \mathrm{and}\quad d_j = 0 \quad [j=2 \ \mathrm{to}\ n] . \end{aligned}$$

(B.2)

The eigenvectors associated with the eigenvalues read

$$\begin{aligned} v_1 = d^{-1/2}_1 x,\quad \mathrm{and}\quad v_{j=2,\ldots,n} \quad\mathrm{orthogonal}\ \mathrm{to}\ v_1 \end{aligned}$$

(B.3)

normalized to unity.

Appendix C: Block-diagonalization: Higgs and vector masses

1. The eigen-masses and mixings in the Higgs sector, when block-diagonalizing the real and symmetric matrix ${\mathcal{M}}^{2} \Rightarrow {\mathcal{M}}^{2}_{m}$ by an orthogonal transformation ${\mathcal{O}}$,

$$\begin{aligned} &{\mathcal{M}^2 = \left ( \begin{array}{c@{\quad}c} M_0^2 & X^T \\ X & M^2 \end{array} \right ) = \left ( \begin{array}{c@{\quad}c} 0 & 0 \\ 0 & M^2 \end{array} \right ) + \left ( \begin{array}{c@{\quad}c} M_0^2 & X^T \\ X & 0 \end{array} \right ) \quad \Rightarrow} \\ &{{ \mathcal{M}}^2_m = {\mathcal{O}} {\mathcal{M}}^2{\mathcal{O}}^T = \left ( \begin{array}{c@{\quad}c} {\hat{M}}_0^2 & 0 \\ 0 & {\hat{M}}^2 \end{array} \right )} \end{aligned}$$

(C.1)

can iteratively be constructed from the lowest order to arbitrary order in the expansion parameter $\epsilon \sim \|M^{2}_{0}\| / \|M^{2}\|, \|X\| / \|M^{2}\|$. The first, large part of the mass matrix will be called ${\mathcal{M}}^{2}_{0}$, the second, small part $\mathcal{E}$ which is order ϵ compared with ${\mathcal{M}}^{2}_{0}$. Thus, the expansion is valid for masses in the hidden sector large compared to SM masses and the mixings.

The conditions which determine the mixing matrix $\mathcal{O}$ for block-diagonalization of the mass matrix $\mathcal{M}^{2}$ are orthogonality and diagonality:

$$\begin{aligned} &{\mathcal{O} = \sum_{N=0}^\infty o_N \quad \mathrm{with} \ o_0 = 1 \ \mathrm{and} \ o_N \sim \epsilon^N,} \end{aligned}$$

(C.2)

$$\begin{aligned} &{{\mathcal{M}}^2_m = \left ( \begin{array}{c@{\quad}c} 0 & 0 \\ 0 & M^2 \end{array} \right ) + \left ( \begin{array}{c@{\quad}c} M_0^2 & 0 \\ 0 & 0 \end{array} \right ) + \sum_{N=2}^\infty \left ( \begin{array}{c@{\quad}c} {\hat{M}}^2_{0,N} & 0 \\ 0 & {\hat{M}}^2_N \end{array} \right ).} \end{aligned}$$

(C.3)

The first two matrices in ${\mathcal{M}}^{2}_{m}$ will occasionally be identified with indices j=0 and 1, respectively.

(i) The orthogonality condition for $\mathcal{O}$ determines the symmetric part of the component o _N from o _j<N as

$$\begin{aligned} o_N + o_N^T = - \sum ^{N-1}_{j=1} o_j o_{N-j}^T . \end{aligned}$$

(C.4)

(ii) The diagonalization condition of the mass matrix determines the antisymmetric part of o _N from the off-diagonal block elements, and at the same time the expansion of the mass eigenvalues from the diagonal block elements:

$$\begin{aligned} &{\left (\begin{array}{c@{\quad}c} {\hat{M}}^2_{0,N} & 0 \\ 0 & {\hat{M}}^2_N \end{array} \right ) = \sum_{j=0}^{N} o_j \mathcal{M}^2_0 o^T_{N-j} +\sum_{j=0}^{N-1} o_j \mathcal{E} o_{N-1-j}^T } \\ &{\quad \mathrm{with}\ {\mathcal{M}}^2_0 = \left (\begin{array}{c@{\quad}c} 0 & 0 \\ 0 & M^2 \end{array} \right ) \ \mathrm{and}\ {\mathcal{E}} = \left (\begin{array}{c@{\quad}c} M^2_0 & X^T \\ X & 0 \end{array} \right )} \end{aligned}$$

(C.5)

for N≥1.

To simplify the notation we switch from indexed symbols to one-letter symbols by denoting

$$ o_N = \left ( \begin{array}{c@{\quad}c} x_N & y_N^T \\ -y_N & z_N \end{array} \right ) . $$

(C.6)

The matrix z=z ^T is taken symmetric, the antisymmetric part of o _N is defined in the y column and row. To unify the mass dimensions and express all the formulas in compact form, we introduce three dimensionless and two dimensionful matrices as

$$ \begin{aligned} &\mu = M^{-2} M^2_0, \qquad y_{(N)+} = M^2 y_{(N)}, \\ &y = M^{-2} X, \qquad y_{(N)-} = M^{-2} y_{(N)}, \\ &\hat{z}_N = M^{-2} z_N M^2 \end{aligned} $$

(C.7)

where y _±, however, always come as dimensionless combinations.

The simplified recurrence relations of the matrix blocks may be cast in the following form for the block-diagonal components:

$$ \begin{aligned} &x_0 = 1\\ &x_1 = 0\\ &x_2 = - \frac{1}{2} y^T y\\ &x_3 = - y^T \mu y \\ &\vdots \\ &x_N = - \frac{1}{2} \sum^{N-1}_{j=1} \bigl( x_j x_{N-j} + y_j^Ty_{N-j} \bigr)\\ &z_0 = 1 \\ &z_1 = 0 \\ &z_2 = -\frac{1}{2} yy^T \\ &z_3 = -\frac{1}{ 2} \{yy^T,\mu\} \\ & \vdots \\ &z_N = -\frac{1}{2} \sum^{N-1}_{j=1} \bigl( z_jz_{N-j} + y_ jy_{N-j}^T \bigr) \end{aligned} $$

(C.8)

and for the off-diagonal components:

$$ \begin{aligned} &y_0 = 0, \qquad y_1 = -y, \qquad y_2 = -\mu y,\\ &y_3 = -\mu^2 y- \frac{1}{2} \bigl(y^T y y - y_+^T y y_-\bigr) \\ &\vdots\\ & y_N = \sum_{j=0}^{N-1} \bigl\{ (\mu y_j - \hat{z}_j y) x_{N-1-j} - \hat{z}_{N-j} y_j \\ &\phantom{y_N =}{}+ y_+^T y_{N-1-j} y_{j-} \bigr\} . \end{aligned} $$

(C.9)

The block-diagonal components of the mass matrix ${\mathcal{M}}^{2}_{m}$ are given by

$$\begin{aligned} &{{\hat{M}}^2_{0,0} = 0, \qquad {\hat{M}}^2_{0,1} = M_0^2,} \\ &{ {\hat{M}}^2_{0,2} = - y^T M^2 y, \qquad {\hat{M}}^2_{0,3} = -M^2_0 y^T y,} \\ &{{\hat{M}}^2_0 = M^2, \qquad {\hat{M}}^2_1 = 0,} \\ &{ {\hat{M}}^2_2 = \frac{1}{2}\{ yy^T, M^2\}, \qquad {\hat{M}}^2_{3} = M^2_0 y y^T,} \\ &{ \vdots}\\ &{ {\hat{M}}^2_{0,N} = \sum_{j=0}^{N} y_j^T M^2 y_{N-j} +\sum _{j=0}^{N-1} \bigl[M^2_0 x_j x_{N-1-j}} \\ &{\phantom{{\hat{M}}^2_{0,N} =}{} + \bigl(y^T_j X + X^T y_j\bigr) x_{N-1-j} \bigr]} \\ &{ {\hat{M}}^2_N = \sum_{j=0}^{N} z_j M^2 z_{N-j} +\sum _{j=0}^{N-1} \bigl[ M^2_0 y_j y^T_{N-1-j} } \\ &{\phantom{{\hat{M}}^2_N =}{}- \bigl(z_j X y^T_{N-1-j} +y_{N-1-j} X^T z_j\bigr) \bigr] .} \end{aligned}$$

(C.10)

2. In the same way the mass matrix in the gauge sector can be diagonalized recursively for small gauge-kinetic mixing s. After applying the KT matrix $\mathcal{Z}$, given in closed form by

$$ {\mathcal{Z}} = \left (\begin{array}{c@{\quad}c@{\quad}c} s_W & c_W & 0 \\ c_W & -s_W & 0 \\ 0 & -\sigma s & \sigma \end{array} \right ) $$

(C.11)

with the symmetric matrix σ=(1−ss ^T)^−1/2, the transformed (2+n)×(2+n) mass matrix

$$\begin{aligned} &{{\mathcal{M}}^2_s = {\mathcal{Z}} { \mathcal{M}}^2_c {\mathcal{Z}}^T = M^2_{Z_c} \left (\begin{array}{c@{\quad}c@{\quad}c} 0 & 0 & 0 \\ 0 & 1 & s_W s^T \sigma \\ 0 & s_W \sigma s & \sigma (\varDelta + s^2_W s s^T) \sigma \end{array} \right )} \\ &{ \quad \mathrm{with} \ \varDelta = M^2_{V_c}/M^2_{Z_c}} \end{aligned}$$

(C.12)

has the characteristic properties which allow the recursive diagonalization according to the algorithm developed in the previous subsection. Disregarding the photonic null-vectors, we can identify, in symbolic notation,

$$ M^2_0 \sim M^2_{Z_c} ; \qquad M^2 \sim M^2_{V_c} ; \qquad X \sim M^2_{Z_c} s $$

(C.13)

for the (1+n)×(1+n) mass submatrix with $\| M^{2} \| \gg M^{2}_{0} \gg \| X \|$. However, the kinetic mass matrix includes additional s-dependent terms which can be expanded for small s. They add contributions $\varSigma {\mathcal{E}}^{k}$ to the matrix $\mathcal{E}$ in Eq. (C.5). Since they affect the matrix o _N only by already known matrices o _j<N−1, they are easy to incorporate. This procedure is straightforward, though technically cumbersome, and we will not present the additional terms in detail.

Appendix D: Re-diagonalization

After the block-diagonalization the mass matrix ${\hat{M}}^{2} = M^{2} + \varDelta$ in the hidden sector is not diagonal anymore. Here, the correction term Δ is of the second order or higher in ϵ. It may be re-diagonalized ${\hat{M}}^{2} \rightarrow [\mathrm{{diag}} {\hat{M^{2}}} ]= M^{2}+\varDelta^{d}$ by the orthogonal transformation U=1+u. Expanding all the matrices, $\varDelta^{(d)} = \sum \varDelta^{(d)}_{N}$ and u=∑u _N , systematically in terms of the power N≥2 of ϵ, the diagonal mass matrix and the orthogonal transformation matrix can easily be constructed recursively, as worked out before.

The orthogonality condition determines the symmetric part of u _N in terms of the predetermined lower-order matrices u _k≤N−2 by the relation

$$ u_N + u_N^T = - \sum ^{N-2}_{k=2} u_k u^T_{N-k}. $$

(D.1)

Introducing the symmetric auxiliary matrix

$$\begin{aligned} A_N =& \sum_{k-2}^{N-2} \Biggl[ u_k M^2 u^T_{N-k} + \bigl(u_{N-k} \varDelta_k + \varDelta_k u^T_{N-k}\bigr) \\ &{} + \sum^{N-2-k}_{l=2} u_k \varDelta_l u^T_{N-k-l} \Biggr] \end{aligned}$$

(D.2)

the diagonalization condition

$$ \varDelta^d_N = \varDelta_N + u_N M^2 + M^2 u^T_N + A_N $$

(D.3)

can be exploited to project out the antisymmetric part of u _N ,

$$\begin{aligned} \bigl[u_N - u_N^T\bigr]_{ab} =& 2 I_{ab} \biggl[ \varDelta_{Nab} + \frac{1}{2} \bigl(M^2_{aa}+M^2_{bb}\bigr) \bigl[u_N+u_N^T\bigr]_{ab} \\ &{} + A_{Nab} \biggr] \quad [a \neq b] \end{aligned}$$

(D.4)

with the abbreviation $I_{ab} = 1 / (M^{2}_{aa} - M^{2}_{bb})$ and the symmetric part calculated before by means of the orthogonality condition. The diagonalized eigenvalues are given by the elements of the matrix $\varDelta^{d}_{N}$ in Eq. (D.2), which, at this point, includes only predetermined matrices u _N ,A _N on the right-hand side. In this way the re-diagonalization of the mass matrix in the hidden sector is completed.

These solutions may be illustrated for the first three non-trivial cases. The second- and third-order terms read

$$\begin{aligned} &{N = {2,3}{:}\quad \varDelta^d_{2,3aa} =\varDelta_{2,3aa},} \end{aligned}$$

(D.5)

$$\begin{aligned} &{\phantom{N = {2,3}{:}\quad}u_{2,3aa} = 0,} \end{aligned}$$

(D.6)

$$\begin{aligned} &{\phantom{N = {2,3}{:}\quad}u_{2,3ab} = I_{ab} \varDelta_{2,3ab} \quad [a\neq b] .} \end{aligned}$$

(D.7)

The transformation matrix u _2,3 is apparently antisymmetric. As a result, the newly diagonalized mass matrix in the hidden sector is found by just truncating the mass matrix after block-diagonalization to the diagonal elements up to the third order. However, starting from the fourth order, there appear non-trivial contributions to the diagonal elements from the lower-order terms. The fourth-order terms read

$$\begin{aligned} &{N = 4{:} \quad \varDelta^d_{4aa} = \varDelta_{4aa} +\sum_{c\neq a} I_{ac} \varDelta^2_{2ac},} \end{aligned}$$

(D.8)

$$\begin{aligned} &{\phantom{N = 4{:} \quad}u_{4aa} = - \frac{1}{2}\sum_{c\neq a} I^2_{ac} \varDelta^2_{2ac},} \end{aligned}$$

(D.9)

$$\begin{aligned} &{\phantom{N = 4{:} \quad}u_{4ab} = I_{ab} \varDelta_{4ab} - I^2_{ab} \varDelta_{2aa} \varDelta_{2ab}} \\ &{\phantom{N = 4{:}\quad u_{4ab} =}{}+ \sum_{c\neq a} I_{ab} I_{ac} \varDelta_{2ac} \varDelta_{2bc} \quad [a\neq b].} \end{aligned}$$

(D.10)

Note that the fourth-order matrix u ₄ is not antisymmetric anymore.